Assignment5

## 
  |                                                                            
  |                                                                      |   0%
  |                                                                            
  |                                                                      |   1%
  |                                                                            
  |=                                                                     |   1%
  |                                                                            
  |=                                                                     |   2%
  |                                                                            
  |==                                                                    |   2%
  |                                                                            
  |==                                                                    |   3%
  |                                                                            
  |==                                                                    |   4%
  |                                                                            
  |===                                                                   |   4%
  |                                                                            
  |===                                                                   |   5%
  |                                                                            
  |====                                                                  |   5%
  |                                                                            
  |====                                                                  |   6%
  |                                                                            
  |=====                                                                 |   6%
  |                                                                            
  |=====                                                                 |   7%
  |                                                                            
  |=====                                                                 |   8%
  |                                                                            
  |======                                                                |   8%
  |                                                                            
  |======                                                                |   9%
  |                                                                            
  |=======                                                               |   9%
  |                                                                            
  |=======                                                               |  10%
  |                                                                            
  |=======                                                               |  11%
  |                                                                            
  |========                                                              |  11%
  |                                                                            
  |========                                                              |  12%
  |                                                                            
  |=========                                                             |  12%
  |                                                                            
  |=========                                                             |  13%
  |                                                                            
  |=========                                                             |  14%
  |                                                                            
  |==========                                                            |  14%
  |                                                                            
  |==========                                                            |  15%
  |                                                                            
  |===========                                                           |  15%
  |                                                                            
  |===========                                                           |  16%
  |                                                                            
  |============                                                          |  16%
  |                                                                            
  |============                                                          |  17%
  |                                                                            
  |============                                                          |  18%
  |                                                                            
  |=============                                                         |  18%
  |                                                                            
  |=============                                                         |  19%
  |                                                                            
  |==============                                                        |  19%
  |                                                                            
  |==============                                                        |  20%
  |                                                                            
  |==============                                                        |  21%
  |                                                                            
  |===============                                                       |  21%
  |                                                                            
  |===============                                                       |  22%
  |                                                                            
  |================                                                      |  22%
  |                                                                            
  |================                                                      |  23%
  |                                                                            
  |================                                                      |  24%
  |                                                                            
  |=================                                                     |  24%
  |                                                                            
  |=================                                                     |  25%
  |                                                                            
  |==================                                                    |  25%
  |                                                                            
  |==================                                                    |  26%
  |                                                                            
  |===================                                                   |  26%
  |                                                                            
  |===================                                                   |  27%
  |                                                                            
  |===================                                                   |  28%
  |                                                                            
  |====================                                                  |  28%
  |                                                                            
  |====================                                                  |  29%
  |                                                                            
  |=====================                                                 |  29%
  |                                                                            
  |=====================                                                 |  30%
  |                                                                            
  |=====================                                                 |  31%
  |                                                                            
  |======================                                                |  31%
  |                                                                            
  |======================                                                |  32%
  |                                                                            
  |=======================                                               |  32%
  |                                                                            
  |=======================                                               |  33%
  |                                                                            
  |=======================                                               |  34%
  |                                                                            
  |========================                                              |  34%
  |                                                                            
  |========================                                              |  35%
  |                                                                            
  |=========================                                             |  35%
  |                                                                            
  |=========================                                             |  36%
  |                                                                            
  |==========================                                            |  36%
  |                                                                            
  |==========================                                            |  37%
  |                                                                            
  |==========================                                            |  38%
  |                                                                            
  |===========================                                           |  38%
  |                                                                            
  |===========================                                           |  39%
  |                                                                            
  |============================                                          |  39%
  |                                                                            
  |============================                                          |  40%
  |                                                                            
  |============================                                          |  41%
  |                                                                            
  |=============================                                         |  41%
  |                                                                            
  |=============================                                         |  42%
  |                                                                            
  |==============================                                        |  42%
  |                                                                            
  |==============================                                        |  43%
  |                                                                            
  |==============================                                        |  44%
  |                                                                            
  |===============================                                       |  44%
  |                                                                            
  |===============================                                       |  45%
  |                                                                            
  |================================                                      |  45%
  |                                                                            
  |================================                                      |  46%
  |                                                                            
  |=================================                                     |  46%
  |                                                                            
  |=================================                                     |  47%
  |                                                                            
  |=================================                                     |  48%
  |                                                                            
  |==================================                                    |  48%
  |                                                                            
  |==================================                                    |  49%
  |                                                                            
  |===================================                                   |  49%
  |                                                                            
  |===================================                                   |  50%
  |                                                                            
  |===================================                                   |  51%
  |                                                                            
  |====================================                                  |  51%
  |                                                                            
  |====================================                                  |  52%
  |                                                                            
  |=====================================                                 |  52%
  |                                                                            
  |=====================================                                 |  53%
  |                                                                            
  |=====================================                                 |  54%
  |                                                                            
  |======================================                                |  54%
  |                                                                            
  |======================================                                |  55%
  |                                                                            
  |=======================================                               |  55%
  |                                                                            
  |=======================================                               |  56%
  |                                                                            
  |========================================                              |  56%
  |                                                                            
  |========================================                              |  57%
  |                                                                            
  |========================================                              |  58%
  |                                                                            
  |=========================================                             |  58%
  |                                                                            
  |=========================================                             |  59%
  |                                                                            
  |==========================================                            |  59%
  |                                                                            
  |==========================================                            |  60%
  |                                                                            
  |==========================================                            |  61%
  |                                                                            
  |===========================================                           |  61%
  |                                                                            
  |===========================================                           |  62%
  |                                                                            
  |============================================                          |  62%
  |                                                                            
  |============================================                          |  63%
  |                                                                            
  |============================================                          |  64%
  |                                                                            
  |=============================================                         |  64%
  |                                                                            
  |=============================================                         |  65%
  |                                                                            
  |==============================================                        |  65%
  |                                                                            
  |==============================================                        |  66%
  |                                                                            
  |===============================================                       |  66%
  |                                                                            
  |===============================================                       |  67%
  |                                                                            
  |===============================================                       |  68%
  |                                                                            
  |================================================                      |  68%
  |                                                                            
  |================================================                      |  69%
  |                                                                            
  |=================================================                     |  69%
  |                                                                            
  |=================================================                     |  70%
  |                                                                            
  |=================================================                     |  71%
  |                                                                            
  |==================================================                    |  71%
  |                                                                            
  |==================================================                    |  72%
  |                                                                            
  |===================================================                   |  72%
  |                                                                            
  |===================================================                   |  73%
  |                                                                            
  |===================================================                   |  74%
  |                                                                            
  |====================================================                  |  74%
  |                                                                            
  |====================================================                  |  75%
  |                                                                            
  |=====================================================                 |  75%
  |                                                                            
  |=====================================================                 |  76%
  |                                                                            
  |======================================================                |  76%
  |                                                                            
  |======================================================                |  77%
  |                                                                            
  |======================================================                |  78%
  |                                                                            
  |=======================================================               |  78%
  |                                                                            
  |=======================================================               |  79%
  |                                                                            
  |========================================================              |  79%
  |                                                                            
  |========================================================              |  80%
  |                                                                            
  |========================================================              |  81%
  |                                                                            
  |=========================================================             |  81%
  |                                                                            
  |=========================================================             |  82%
  |                                                                            
  |==========================================================            |  82%
  |                                                                            
  |==========================================================            |  83%
  |                                                                            
  |==========================================================            |  84%
  |                                                                            
  |===========================================================           |  84%
  |                                                                            
  |===========================================================           |  85%
  |                                                                            
  |============================================================          |  85%
  |                                                                            
  |============================================================          |  86%
  |                                                                            
  |=============================================================         |  86%
  |                                                                            
  |=============================================================         |  87%
  |                                                                            
  |=============================================================         |  88%
  |                                                                            
  |==============================================================        |  88%
  |                                                                            
  |==============================================================        |  89%
  |                                                                            
  |===============================================================       |  90%
  |                                                                            
  |===============================================================       |  91%
  |                                                                            
  |================================================================      |  91%
  |                                                                            
  |================================================================      |  92%
  |                                                                            
  |=================================================================     |  92%
  |                                                                            
  |=================================================================     |  93%
  |                                                                            
  |=================================================================     |  94%
  |                                                                            
  |==================================================================    |  94%
  |                                                                            
  |==================================================================    |  95%
  |                                                                            
  |===================================================================   |  95%
  |                                                                            
  |===================================================================   |  96%
  |                                                                            
  |====================================================================  |  96%
  |                                                                            
  |====================================================================  |  97%
  |                                                                            
  |====================================================================  |  98%
  |                                                                            
  |===================================================================== |  98%
  |                                                                            
  |===================================================================== |  99%
  |                                                                            
  |======================================================================|  99%
  |                                                                            
  |======================================================================| 100%

The graphs you will see below use CalEnviroScreen 4.0 Data which refers to the year of 2020.

There is a total of 21 indicators, including Asthma and PM2.5 which we will closely look at. “Each indicator is assigned a score for each census tract in the state based on the most up-to-date suitable data. Scores are weighted and added together within the two groups to derive a pollution burden score and a population characteristics score. Those scores are multiplied to give the final CalEnviroScreen score.”

Asthma Indicator: spatially modeled, age-adjusted rate of ED visits for asthma per 10,000

PM 2.5 Indicator: annual mean concentration of PM2.5 (weighted average of measured monitor concentrations and satellite observations, μg/m3)

The first map graphically displays Asthma indicator levels throughout the Bay Area while the second graph does the same for PM2.5 concentration as a means of measuring air quality.

This scatter plot shows the correlation between the values for the Asthma Indicator values and the PM2.5 concentration in Air Quality. Although the best fit line shows a positive correlation at this point, it does not appear to be a great predictor of the data. Especially for PM2.5 values between 8 and 9. Ultimately, I predict the residuals will be larger than we would like them to ideally be.

Here is a summary of the model:

## 
## Call:
## lm(formula = Asthma ~ PM2.5, data = bay_asthma_pm_tract)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -54.47 -25.89  -9.61  12.94 182.95 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -116.278     13.040  -8.917   <2e-16 ***
## PM2.5         19.862      1.534  12.950   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 37.49 on 1578 degrees of freedom
## Multiple R-squared:  0.09606,    Adjusted R-squared:  0.09549 
## F-statistic: 167.7 on 1 and 1578 DF,  p-value: < 2.2e-16

As you can see, “An increase of PM2.5 in one unit is associated with an increase of Asthma in 19.862”; “9.6% of the variation in PM2.5 is explained by the variation in Asthma”.

From this plot, you can observe the highest density of residuals is around -20, showing how innacurate the line of best fit was.

Now, we can see the same data layered with a log function. The correlation is clearly significantly stronger. The line of best fit in the scatter plot appears to be way more representative which is confirmed by the data in the table (reduction from 19.862 to 0.35633) and the residuals graph.

## 
## Call:
## lm(formula = log(Asthma) ~ PM2.5, data = bay_asthma_pm_tract)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.00402 -0.46479  0.03313  0.42298  1.75525 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.69234    0.22840   3.031  0.00248 ** 
## PM2.5        0.35633    0.02686  13.264  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6566 on 1578 degrees of freedom
## Multiple R-squared:  0.1003, Adjusted R-squared:  0.09974 
## F-statistic: 175.9 on 1 and 1578 DF,  p-value: < 2.2e-16

This plot of the residuals of the log function has it’s highest density concentrated around 0 showing the accuracy of this new model. Compared to the same graph of the original data, there is a visible shift to the right.

Combining the residuals with spatial information, we can see how the accuracy of the model varies according to location. From my understanding, a low residual represents a high accuracy of the modelled regression and the data collected, meaning it could be used to make reasonable estimates. A positive residual represents an overestimation while a negative value represents an underestimation.

As seen in the map, the area with lowest residual is Stanford’s campus with a -2.004 value.